Darboux cyclides

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Overview

We show animations of Darboux cyclides together with some of their inversions.

A surface X is of celestial type (c,d,n) if

The singular type (A,B) is a pair of Dynkin types where A denotes the real structure and B denotes the type of the isolated singularities. The underlined Dynkin types correspond to real isolated singularities. See here for more information. For the names of the cyclides we use the following abbreviations:

S = sphere, N = non-rotational,
E = elliptic/ellipsoid, C = circular,
H = hyperbolic/hyperboloid, P = parabolic/paraboloid,
1 = of one sheet, 2 = of two sheets,
O = cone, Y = cylinder.

A Darboux cyclide is either one of the following:

2S cyclide HP cyclide
1S cyclide CH1 cyclide
EH2 cyclide spindle cyclide
NE cyclide ring cyclide
EP cyclide Perseus cyclide
Blum cyclide EY cyclide
EH1 cyclide horn cyclide
EO cyclide

The images were made using Surfex.

2S cyclide

Celestial type: (2,4,3)

Singular type: (D4,A0)

Picture Picture Picture
Picture Picture Picture

1S cyclide

Celestial type: (2,4,3)

Singular type: (3A1,A0)

Picture Picture Picture
Picture Picture Picture

EH2 cyclide

Celestial type: (2,4,3)

Singular type: (3A1,A1)

Picture Picture Picture
Picture Picture Picture

NE cyclide

Celestial type: (2,4,3)

Singular type: (3A1,A1)

Picture Picture Picture
Picture Picture Picture

EP cyclide

Celestial type: (2,4,3)

Singular type: (3A1,A2)

Picture Picture Picture
Picture Picture Picture
Picture

Blum cyclide

Celestial type: (6,4,3)

Singular type: (2A1,A0)

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EH1 cyclide

Celestial type: (4,4,3)

Singular type: (2A1,A1)

Picture Picture Picture Picture
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EO cyclide

Celestial type: (3,4,3)

Singular type: (2A1,2A1)

Picture Picture Picture
Picture Picture Picture

HP cyclide

Celestial type: (2,4,3)

Singular type: (2A1,A2)

Picture Picture Picture
Picture Picture Picture

CH1 cyclide

Celestial type: (3,4,3)

Singular type: (2A1,A1+2A1)

Picture Picture Picture
Picture Picture Picture
Picture

spindle cyclide

Celestial type: (3,4,3)

Singular type: (2A1,2A1+2A1)

a.k.a. CO cyclide

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ring cyclide

Celestial type: (4,4,3)

Singular type: (2A1,4A1)

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Perseus cyclide

Celestial type: (5,4,3)

Singular type: (2A1,2A1)

Picture Picture Picture
Picture Picture Picture

EY cyclide

Celestial type: (3,4,3)

Singular type: (2A1,A3)

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Picture

horn cyclide

Celestial type: (3,4,3)

Singular type: (2A1,A3+2A1)

a.k.a. CY cyclide

Picture Picture Picture
Picture Picture Picture
Picture

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